Well-posedness for the fourth-order nonlinear Schrödinger-type equation related to the vortex filament

Abstract

We consider the time-local well-posedness for the initial-value problem of the fourth-order nonlinear Schrödinger-type equation in one space dimension which describes the motion of the vortex filament. By using the method of Fourier restriction norm introduced by Bourgain [3] and Kenig-Ponce-Vega [17]--[19], we show the time-local well-posedness in the Sobolev space $H^s(\mathbb R)$ with $s\ge1/2$ under certain coefficient conditions.